Morris Ang
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View article: Radial conformal welding in Liouville quantum gravity
Radial conformal welding in Liouville quantum gravity Open
The seminal work of Sheffield showed that when random surfaces called Liouville quantum gravity (LQG) are conformally welded, the resulting interface is Schramm-Loewner evolution (SLE). This has been proved for a variety of configurations,…
View article: SLE Loop Measure and Liouville Quantum Gravity
SLE Loop Measure and Liouville Quantum Gravity Open
As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQ…
View article: Boundary touching probability and nested-path exponent for non-simple CLE
Boundary touching probability and nested-path exponent for non-simple CLE Open
The conformal loop ensemble (CLE) has two phases: for $κ\in (8/3, 4]$, the loops are simple and do not touch each other or the boundary; for $κ\in (4,8)$, the loops are non-simple and may touch each other and the boundary. For $κ\in(4,8)$,…
View article: FZZ formula of boundary Liouville CFT via conformal welding
FZZ formula of boundary Liouville CFT via conformal welding Open
Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) pro…
View article: Conformal welding of quantum disks and multiple SLE: the non-simple case
Conformal welding of quantum disks and multiple SLE: the non-simple case Open
Two-pointed quantum disks with a weight parameter $W>0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We extend the conformal welding of quantum disks in [AHS23] to the non-simple regime, and give a c…
View article: Cutting $γ$-Liouville quantum gravity by Schramm-Loewner evolution for $κ\not\in \{γ^2, 16/γ^2\}$
Cutting $γ$-Liouville quantum gravity by Schramm-Loewner evolution for $κ\not\in \{γ^2, 16/γ^2\}$ Open
There are many deep and useful theorems relating Schramm-Loewner evolution (SLE$_κ$) and Liouville quantum gravity ($γ$-LQG) in the case when the parameters satisfy $κ\in \{γ^2, 16/γ^2\}$. Roughly speaking, these theorems say that the SLE$…
View article: Reversibility of whole-plane SLE for $κ> 8$
Reversibility of whole-plane SLE for $κ> 8$ Open
Whole-plane SLE$_κ$ is a random fractal curve between two points on the Riemann sphere. Zhan established for $κ\leq 4$ that whole-plane SLE$_κ$ is reversible, meaning invariant in law under conformal automorphisms swapping its endpoints. M…
View article: Supercritical Liouville quantum gravity and CLE$_4$
Supercritical Liouville quantum gravity and CLE$_4$ Open
We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,…
View article: Critical Liouville quantum gravity and CLE$_4$
Critical Liouville quantum gravity and CLE$_4$ Open
Consider a critical ($γ=2$) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter $κ=4$. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE$_4$ lo…
View article: Derivation of all structure constants for boundary Liouville CFT
Derivation of all structure constants for boundary Liouville CFT Open
We prove that the probabilistic definition of the most general boundary three-point and bulk-boundary structure constants in Liouville conformal field theory (LCFT) agree respectively with the formula proposed by Ponsot-Techsner (2002) and…
View article: Liouville conformal field theory and the quantum zipper
Liouville conformal field theory and the quantum zipper Open
Sheffield showed that conformally welding a $γ$-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter $κ= γ^2$ as the interface, and Duplantier-Miller-Sheffield proved similar result…
View article: The SLE loop via conformal welding of quantum disks
The SLE loop via conformal welding of quantum disks Open
We prove that the SLEκ loop measure arises naturally from the conformal welding of two γ-Liouville quantum gravity (LQG) disks for γ2=κ∈(0,4). The proof relies on our companion work on conformal welding of LQG disks and uses as an essentia…
View article: Conformal welding of quantum disks
Conformal welding of quantum disks Open
Two-pointed quantum disks with a weight parameter W>0 are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their bounda…
View article: Quantum triangles and imaginary geometry flow lines
Quantum triangles and imaginary geometry flow lines Open
We define a three-parameter family of random surfaces in Liouville quantum gravity (LQG) which can be viewed as the quantum version of triangles. These quantum triangles are natural in two senses. First, by our definition they produce the …
View article: The SLE loop via conformal welding of quantum disks
The SLE loop via conformal welding of quantum disks Open
We prove that the SLE$_κ$ loop measure arises naturally from the conformal welding of two $γ$-Liouville quantum gravity (LQG) disks for $γ^2 = κ\in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses a…
View article: The moduli of annuli in random conformal geometry
The moduli of annuli in random conformal geometry Open
We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled pl…
View article: Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond
Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond Open
The scaling limit of the probability that $n$ points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory (CFT). Although this is not fully understood, Delfino and Viti (2010) made a re…
View article: Integrability of SLE via conformal welding of random surfaces
Integrability of SLE via conformal welding of random surfaces Open
We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact fo…
View article: FZZ formula of boundary Liouville CFT via conformal welding
FZZ formula of boundary Liouville CFT via conformal welding Open
Liouville Conformal Field Theory (LCFT) on the disk describes the conformal factor of the quantum disk, which is the natural random surface in Liouville quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov (2000) pro…
View article: Conformal welding of quantum disks
Conformal welding of quantum disks Open
Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their bo…
View article: Brownian loops and the central charge of a Liouville random surface
Brownian loops and the central charge of a Liouville random surface Open
We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appr…
View article: Volume of metric balls in Liouville quantum gravity
Volume of metric balls in Liouville quantum gravity Open
We study the volume of metric balls in Liouville quantum gravity (LQG). For $γ\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \…
View article: Volume of metric balls in Liouville quantum gravity
Volume of metric balls in Liouville quantum gravity Open
We study the volume of metric balls in Liouville quantum gravity (LQG). For $\\gamma \\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly …
View article: Comparison of discrete and continuum Liouville first passage percolation
Comparison of discrete and continuum Liouville first passage percolation Open
Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $γ$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle…
View article: Liouville quantum gravity surfaces with boundary as matings of trees
Liouville quantum gravity surfaces with boundary as matings of trees Open
For $γ\in (0,2)$, the quantum disk and $γ$-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with bou…
View article: Comparison of discrete and continuum Liouville first passage percolation
Comparison of discrete and continuum Liouville first passage percolation Open
Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the $\\gamma $-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle aver…